Suppose that $M$ is a compact manifold with corners, where each boundary hypersurface is an embedded submanifold. Then, do we have an integration by parts identity? i.e.
\begin{align*}
\int_M g(\nabla f, X) \,d\textrm{vol} = \int_{\partial M} f\left\langle X,N \right\rangle \,d\textrm{vol}_{\tilde{g}} - \int_M f\cdot (\operatorname{div} X ) \,d\textrm{vol}
\end{align*}
with $f\in C^\infty(M), X\in \Gamma(M,TM)$. I know that this identity hold for domains with lipschitz boundary, but it is not very clear to me if a domain with corners is a special case of a lipschitz domain.